Surface Tension

Surface tension is basically a property of liquid. The liquid surface behaves like a stretched elastic membrane which has a natural tendency to contract and tends to have a minimum possible area. This property of liquid is called surface tension. INTERMOLECULAR FORCES l. The force which acts between the atoms or the molecules of different substances is called intermolecular force. This force is of two types. (a) Cohesive force – The force acting between the molecules of one type of molecules of same substance is called cohesive force. (b) Adhesive force – The force acting between different types of molecules or molecules of different substance is called adhesive force. 2. Intermolecular forces are different from the gravitational forces not obey the inverse–square law 3. The distance upto which these forces effective, is called molecular range. This distance is nearly 10–9 m. Within this limit this increases very rapidly as the distance decreases. 4. Molecular range depends on the nature of the substance Examples : (a) Water Glass : Water wets glass surface but mercury does not. Because when water comes in contact with glass the adhesive force acts between water and glass molecules. As adhesive force is greater than the cohesive force of water molecules, the water molecules, cling with glass surface and surface become wet. In case of mercury adhesive force is less than that of cohesive force and Hg–molecules do not cling with glass surface and surface does not wet with Hg–molecules. (b) Oil–water : Since Cohesive force of water >Adhesive force oil–water> Cohesive force of oil. (i) If water drop is poured on the surface of oil, it contracts in the form of globule. (ii) If oil drop is poured on the surface of water it spreads to a larger area in the form of thin film. (c) Ink–paper : Since adhesive force between ink–paper > cohesive force on ink, the ink sticks to the paper. Explanation of Surface Tension (Molecular theory of surface Tension) Laplace firstly explained the phenomenon of surface tension on the basis of intermolecular forces. According to him surface tension is a molecular phenomenon and its root causes are electromagnetic forces. He explained the cause of surface tension as described under. If the distance between two molecules is less than the molecular range C (  10–9 m) then they attract each other, but if the distance is more than this then the attraction becomes negligible. If a sphere of radius C with a molecule at centre is drawn, then only those molecules which are enclosed within this sphere can attract or be attracted by the molecule at the centre of the sphere. This sphere is called sphere of molecular activity or sphere of influence. In order to understand the tension acting at the free surface of liquid, let us consider four liquid molecules like A, B, C and D along with their spheres of molecular activity. (a) According to figure D sphere is completely inside liquid. So molecule is attracted equally in all directions and hence resultant force is equal to zero. (b) According to figure sphere of molecule C is just below the liquid surface. So resultant force is equal to zero. (c) The molecule B which is a little below the liquid surface is more attracted downwards due to excess of molecules downwards. Hence the resultant force is acting downwards. MECHANICAL PROPERTIES OF MATTER (d) Molecule A is situated at surface so that its sphere of molecular activity is half outside the liquid and half inside. Only down portion has liquid molecules. Hence it experiences a maximum downward force. Thus all the molecules situated between the surface and a plane XY, distant C below the surface, experience a resultant downward cohesive force. When the surface area of liquid is increased molecules from the interior of the liquid rise to the surface. A X B C net downward force D liquid surface P C Y zero resultant force As these molecules reach near the surface, work is done against the downward cohesive force. This work is stored in the molecules in the form of potential energy. Thus the potential energy of the molecules lying in the surface is greater than that of the molecules in the interior of the liquid. A system is in stable equilibrium when its potential energy is minimum. Hence in order to have minimum potential energy the liquid surface tends to have minimum number of molecules in it. In other words the surfaces tends to contract to a minimum possible area. This tendency is exhibited as surface tension. (i) Surface tension is a scalar quantity. (ii) It acts tangential to liquid surface. (iii) Surface tension is always produced due to cohesive force. (iv) More is the cohesive force, more is the surface tension. (v) When surface area of liquid is increased, molecules from the interior of the liquid rise to the surface. For this, work is done against the downward cohesive force. As a result, its potential energy increases and internal energy decreases. So on increase in surface area cooling occurs. If liquid temperature remains same, then extra energy may be given by external agency. So the molecules in the surface have some additional energy due to their position. This additional energy per unit area of the surface is called Surface Energy, free surface energy or surface energy density. DEPENDENCY OF SURFACE TENSION l On Cohesive Force Those factors which increase the cohesive force between molecules increase the surface tension and those which decrease the cohesive force between molecules decrease the surface tension. 2 On Impurities If the impurity is completely soluble then on mixing it in the liquid, its surface tension increases. e.g., on dissolving ionic salts in small quantities in a liquid, its surface tension increases. If the impurity is partially soluble in a liquid then its surface tension decreases because adhesive force between insoluble impurity molecules and liquid molecules decreases cohesive force effectively, e.g. (a) On mixing detergent in water its surface tension decreases. (b) Surface tension of water is more than (alcohol + water) mixture. 3 On Temperature On increasing temperature surface tension decreases. At critical temperature and boiling point it becomes zero. Note : Surface tension of water is maximum at 4°C 4 5 On Contamination The dust particles or lubricating materials on the liquid surface decreases its surface tension. On Electrification DEFINITION OF SURFACE TENSION Surface tension can be defined in the form of an imaginary line on the surface or by relating it to the work done. The force acting per unit length of an imaginary line drawn on the free liquid surface at right angles to the line and in the plane of liquid surface, is defined as surface tension. Let an imaginary line AB be drawn in any direction on a liquid surface. The surface on either side of this line exerts a pulling force, which is perpendicular to line AB. If force is F and length of AB is L then T = SI Units : N/m and J/m2 CGS Units : dyne/cm and erg/cm2 F B F A F L Dimensions : M1L0 T–2 Examples : (i) When any needle floats on the liquid surface then 2T sin = mg Ex. A mosquito sitting on a liquid surface. (ii) If the needle is lifted from the liquid surface then required excess F force will be Fexcess = 2T Minimum force required Fmin = mg + 2T (iii) T T Required excess force for a circular thick ring (or hollow disc) having internal and external radii r1 and r2 is dipped in and taken out from liquid Fexcess = F1 + F2 = T(2r1) + T(2r2) = 2T(r1 + r2) (iv) Required excess force for a circular ring (r1 = r2 = r) Fexcess = 2T( r + r) = 4 r T T (v) Surface Energy W r Required excess force for a circular disc (r1 = 0, r2 = r) Fexcess = 2 rT T T According to molecular theory of surface tension the molecule in the surface have some additional energy due to their position. This additional energy per unit area of the surface is called ' Surface energy". Let a liquid film be formed on a wire frame and a straight wire of length  can slide on this wire frame as shown in figure. The film has two surfaces and both the surfaces are in contact with the sliding wire and hence, exert forces of surface tension on it. If T be the surface tension of the solution, each surface will pull the wire parallel to itself with a force T. MECHANICAL PROPERTIES OF MATTER Thus, net force on the wire due to both the surfaces is 2T. Apply an external force F equal and opposite to it to keep the wire is equilibrium. Thus, F=2T Now, suppose the wire is moved through a small distance dx, the work done by the force is, dW = F dx = (2T) dx But (2) (dx) is the total increase in area of both the surfaces of the film. Let it be dA. Then, dW = T dA T = dW/dA Thus, the surface tension T can also be defined as the work done in increasing the surface area by unity. Further, since there is no change in kinetic energy, the work done by the external force is stored as the potential energy of the new surface. T= dU [as dW = dU] dA Special Cases (i) Work done (surface energy) in formation of a drop of radius r = Work done against surface tension W = Surface tension T × change in area A = T × 4r2 = 4r2T (ii) Work done (surface energy) in formation of a soap bubble of radius r : W = T × A or W = T × 2 × 4r2 = 8r2T [ soap bubble has two surfaces] Ex. The length of a needle floating on water is 2.5 cm. Calculate the added force required to pull the needle out of water. [T = 7.2 × 10–2 N/m] Sol. The force of surface tension F = T× 2 ( Two free surfaces are there) F = 7.2 × 10–2 × 2 × 2.5 × 10–2 = 3.6 × 10–3 N Ex. A paper disc of radius R from which a hole of radius r is cut out, is floating in a liquid of surface tension, T. What will be force on the disc due to surface tension? Sol. T= Ex. Calculate the work done against surface tension in blowing a soap bubble from a radius 10 cm to 20 cm if the surface tension of soap solution is 25 × 10–3 N/m. Then compare it with liquid drop of same radius. Sol. (i) For soap bubble : Extension in area = 2 × 4r22 –2 × 4r12 =8 [(0.2)2– (0.1)2] = 0.24  m2 F F =  F = 2 (R + r)T 2  (R  r) L Work done W1 = surface tension × extension in area = 25 × 10–3 × 0.24  = 6 × 10–3 J (ii) For Liquid Drop : in case of liquid drop only one free surface, so extension in area will be half of soap bubble  W2 = W1 = 3 × 10–3 J 2 SPLITTING OF BIGGER DROP INTO SMALLER DROPLETS If bigger drop is spitted into smaller droplets then in this process volume of liquid always remain conserved. Let bigger drop has radius R. It is splitted into n smaller drops of radius r then by conservation of volume 3 (i) (ii) 4 4  R  R R 3  n  r 3   n     r = 1 / 3 3 n 3  r  Initial surface area = 4R2 and final surface area = n(4r2) Therefore initial surface energy Ei = 4R2T and final surface energy Ef = n(4r2T) Change in area A = n4r2 – 4R2 = 4(nr2 – R2) Therefore the amount of surface energy absorbed i.e. E = Ef – Ei = 4T (nr2 – R2)  Magnitude of work done against surface tension i.e. W = 4(nr2 – R2)T R  W = 4T (nr2 – R2) = 4R2 T (n1/3 – 1) = 4R2T   1   W = 4R3T r  1 1  r  R   In this process temperature of system decreases as energy gets absorbed in increasing surface area. 4  R 3 T 1 1  3 T  1 1  1 1     W = J m s  = 4R T   r  R  = J s  r  R    4    r R  R3 J  s  3 3 Where  = liquid density, s = liquid's specific heat Thus in this process area increasing, surface energy increasing, internal energy decreasing, temperature decreasing, and energy is absorbed. Ex. A big drop is formed by coalescing 1000 small droplets of water .What will be the change in surface energy. What will be the ratio between the total surface energy of the droplets and the surface energy of the big drop ? Sol. By conservation of volume 4 4 R3 = 1000 × r3 3 3 r = R 10 2 R Surface energy of 1000 droplets = 1000 × T × 4   = 10 (T × 4R2)  10  Surface energy of the big drop = T × 4R2 Surface energy will decrease in the process of formation of bigger drop, hence energy is released and temperature total surface energy of 100 0 droplets 1 0(T  4 R 2 ) 10  increases .  = surface energy of big drop 1 T  4 R 2 Ex. A water drop of radius 1mm is broken into 10 6 identical drops. Surface tension of water is 72 dynes/cm. Find the energy spent in this process. Sol. As volume of water remains constant, so 4 4 R R 3 = n r3 r = 1 3 3 3 n Increase in surface area A = n (4r2) – 4R2 = 4 (n1/3 – 1) R2 = 4 (100 – 1)10–6  Energy spent = TA = 4 × 99 × 10–6 × 72 × 10–3 = 89.5 × 10–6 J EXCESS PRESSURE INSIDE A CURVED LIQUID SURFACE The pressure on the concave side of curved liquid surface is greater than that on the convex side. Therefore pressure difference exists across two sides of a curved surface. This pressure difference is called excess pressure. Excess pressure inside the drop Let a drop of radius r having internal and external pressure Pi and P0 respectively, so that excess pressure Pex =(Pi – P0). dr If the radius of the drop is changed from r to (r+dr) then Work done = F.dr = (PA) dr = P 4r2dr Change in surface area = r(r + dr)2–4r2= 8rdr So by definition of surface energy T = P0 Pi r W 4 r 2 Pdr 2T = Pex =  Pi  P0  = A r 8 rdr Excess pressure inside soap bubble : Since the soap bubble has two surfaces. The excess pressure will get doubled as compared to a drop Pi – P' = 2T 2T 4T , P' – P0 =  excess pressure = Pi – P0 = r r r r Pi P0 P’ MECHANICAL PROPERTIES OF MATTER (i) For liquid surface, pressure on concave side is always higher than convex side . low high high high low (ii) If a bubble is formed inside a liquid, the pressure inside the bubble is more than the pressure outside the bubble. (iii) A soap film is formed in a wire frame. A loop of thread is lying on the film. If the film inside the loop is broken then the tension in the thread will be 2Tr. (iv) In the following arrangement, air will flow from bubble A to B if T 2 and T3 are opened, because pressure in A is greater than in B. T2 A T1 T3 B 2 AT d (v) The force required to separate two plates of Area A is given by F = Ex. Prove that If two bubbles of radii r1 and r2 coalesce isothermally in vacuum then the radius of new bubble will be r  r12  r22 Sol. When two bubbles coalesce then number of molecules of air will remain constant and temperature also constant so 4T  4 3 4T  4 3 4T  4 3 n1 + n2 = n P1V1 + P2V2 = PV  r  3 r1   r  r2  =  r  r= 3 r 3 1 2 Ex. r12  r22 Prove that If two bubbles of radii r1 and r2 (r1 < r2) come in contact with each other then the radius of curvature of the r1 r2 common surface r = r  r . 2 1 Sol.  r1 < r2  P1 > P2 Small part of bubbles is in equilibrium r1 P1  A  – P2  A  = Ex. r2 P1 P2 r1 r2 4T 4T 4T 4T A  – = r = r r1 r2 r r 2  r1 Calculate the excess pressure within a bubble of air of radius 0.1 mm in water. If the bubble had been formed 10 cm below the water surface on a day when the atmospheric pressure was 1.013 × 105 Pa, then what would have been the total pressure inside the bubble? Surface tension of water = 73 × 10–3 N/m Sol. Excess pressure Pexcess = 2 T 2  7 3  1 0 3 = =1460 Pa r 0 .1  10 3 The pressure at a depth d, in liquid is P = hdg. Therefore, the total pressure inside the air bubble is Pin = Patm + hdg + 2T = (1.013 × 105) + (10 × 10–2 × 103 × 9.8) + 1460 r = 101300 + 980 + 1460 = 103740 = 1.037 × 105 Pa Ex. The limbs of a manometer consist of uniform capillary tubes of radii are 10–3 & 7.2 × 10–4 m. Find out the correct pressure difference if the level of the liquid in narrower tube stands 0.2 m above that in the broader tube.(Density of liquid =103 kg/m3, Surface tension = 72 × 10–3 N/m) Sol. If P1 and P2 are the pressures in the broader and narrower tubes of radii r1 and r2 respectively, the pressure just below 2T 2T the meniscus in the respective tubes will be P1– r and P2– r2 1  2T   2T  So that P1  r  – P2  = h g or P1–P2 = hg –2T r2   1   1 1      r2 r1  1 1    P1 – P2 = 0.2 × 103 × 9.8 –2 × 72 × 10–3  4 4  = 1960– 97 = 1863 Pa 14  1 0   7.2  10 ANGLE OF CONTACT (C) The angle enclosed between the tangent plane at the liquid surface and the tangent plane at the solid surface at the point of contact inside the liquid is defined as the angle of contact. The angle of contact depends the nature of the solid and liquid in contact. 1 Effect of Temperature on angle of contact 1  On increasing temperature surface tension decreases, thus cosc increases Q cos c   and cdecrease. T  So on increasing temperature, c decreases. 2 Effect of Impurities on angle of contact (a) Solute impurities increase surface tension, so cosc decreases and angle of contact c increases. (b) Partially solute impurities decrease surface tension, so angle of contact cdecreases. 3 Effect of Water Proofing Agent Angle of contact increases due to water proofing agent. It gets converted acute to obtuse angle. Table of angle of contact of various solid–liquid pairs S o lid - Liquid Pair ° C Glass -Normal water Glass -Distilled water Glass - Alcohol Glass - Mercury Paraffin wax - Water Silver - Water 8° 0° 0° 135° 108° 90° MECHANICAL PROPERTIES OF MATTER Shape of Liquid Surface When a liquid is brought in contact with a solid surface, the surface of the liquid becomes curved near the place of contact. The shape of the surface (concave or convex) depends upon the relative magnitudes of the cohesive force between the liquid molecules and the adhesive force between the molecules of the liquid and the solid. The free surface of a liquid which is near the walls of a vessel and which is curved because of surface tension is known as meniscus. The cohesive force acts at an angle 45° from liquid surface whereas the adhesive force acts at right angles to the solid surface. The relation between the shape of liquid surface, cohesive/adhesive forces, angle of contact, etc are summarised in the table below : Relation Between Cohesive and Adhesive Force FA concave surface FR FC FA horizontal surface FR water FC 2 FA FC water FC mercury FR silver glass FA  convex surface FA  glass FC FA  2 FC 2 Shape of meniscus Concave Plane Convex Angle of contact C < 90° (Acute angle) C = 90° (Right angle) C > 90° (Obtuse angle) Shape of liquid drop C Level of liquid Liquid rises up nor falls Liquid neither rises Liquid falls Wetting property Liquid wets the Liquid does not wet Liquid does not wet solid surface the solid surface the solid surface Glass – Water Silver – Water Example Glass – Mercury CAPILLARY TUBE AND CAPILLARITY A glass tube with fine bore and open at both ends is known as capillary tube. The property by virtue of which a liquid rise or depress in a capillary tube is known as capillarity. Rise or fall of liquid in tubes of narrow bore (capillary tube) is called capillary action. Calculation of Capillary Rise When a capillary tube is first dipped in a liquid as shown in the figure, the liquid climbs up the walls curving the surface. The liquid continues to rise in the capillary tube until the weight of the liquid column becomes equal to force due to surface tension. Let the radius of the meniscus is R and the radius of the capillary tube is r. The angle of contact is surface tension is T, density of liquid is  and the liquid rises to a height h. Now let us consider two points A and B at the same horizontal level as shown. By Pascal's law PA = PB  PA = PC + gh PA – PC = gh  (PB = PC + gh) Now, the point C is on the curved meniscus which has PA and PC as the two pressures on its concave and convex sides respectively.  Zurin's Law : PA - PC = 2T 2T 2T 2Tcosθ = = ρ gh  2Tcosθ = rρ gh  h =  R r / cosθ r / cosθ rρg The height of rise of liquid in a capillary tube is inversely proportional to the radius of the capillary tube, if T,  and g are constant h  versa. 1 or rh = constant. It implies that liquid will rise more in capillary tube of less radius and vice r (i) For water and clean glass capillary c (ii) If angle of contact c is acute then cosc is positive, so h is positive and liquid rises. If c is obtuse then cosc is negative, so h is negative, therefore liquid depresses. (iii) Rise of liquid in a capillary tube does not obey the law of conservation of energy. (iv) For two capillaries of radii r1 and r2, the capillary rise h1 and h2 are such that and r is graphically represented as (v) h 2 r1  h 1 r2 . The relation between h h r Inside a satellite, water will rise upto top level but will not come outside. Radius of curvature(R') increases hR . (It is in accordance with Zurin's law). R' (vi) If a capillary tube is dipped into a liquid and tilted at an angle from vertical then the vertical height of liquid column remains same whereas the length of liquid column in the capillary tube increases. in such a way that final height h' is reduced and given by h '  h = cos  l  (v) h cos  The height 'h' is measured from the bottom of the meniscus. However, there exist some liquid above this line also. If 1   rg h  r  3   correction of this is applied then the formula will be T  2 cos  (vi) If a hollow sphere of radius r which has a fine hole, drowned in a vessel upto h depth, then liquid will not enter upto critical height h, given by hg= 2 T cos  r MECHANICAL PROPERTIES OF MATTER Ex. Calculate the height to which water will rise in a capillary tube of diameter 1 × 10–3 m. [Given : surface tension of water is 0.072 N m–1, angle of contact is 0°, g = 9.8 m s–2 and density of water = 1000 kg m–3] Sol. Height of capillary rise h  Ex. Water rises to a height of 20 mm in a capillary. If the radius of the capillary is made one third of its previous value then what is the new value of the capillary rise? Sol. Since h   2 T cos  2  0.072  cos 0 0  m  2 .94  10 2 m r g 5  10 4  1 00 0  9 .8 2 Tcos and for the same liquid and capillaries of difference radii h1r1 = h2r2 rdg h 2 r1 1 = = = 3 hence h2 = 3h1 = 3 × 20 mm = 60 mm h1 r2 1 / 3 20